As in [1]}, the properties of \(\widetilde{T}_{\mathrm {c}}\)  and \(\widetilde{\Omega }_{\mathrm {c}}\)  are readily checked.
As for the relation \({\hbox{\rm Ad\,}}\widetilde{T}_{\mathrm {c}}(a) (\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }) \subset \widetilde{\mathcal {M}}_{\mathrm {c},\kappa }\)  for \(a \in W_{\mathrm {R}}\) , one can prove this by noting
that \(\widetilde{V}_{\mathrm {c},\kappa }\)  and \(\widetilde{T}_{\mathrm {c}}(a)\)  commute and the assumptions that \(\mathcal {M}_{\mathrm {c}}\)  and \(T_{\mathrm {c}}\)  have the relation.
